Maximum likelihood estimation in a class of nonregular cases

Abstract

We consider maximum likelihood estimation of the parameters of a probability density which is zero for x < θ and asymptotically αc(x-θ)^α−1 as x ↓ θ. Here θ and other parameters, which may or may not include α and c₁ are unknown. The classical regularity conditions for the asymptotic properties of maximum likelihood estimators are not satisfied but it is shown that, when α> 2, the information matrix is finite and the classical asymptotic properties continue to hold. For α= 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For 1 < α < 2, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For αα 1, the maximum likelihood estimators may not exist at all, but alternatives are proposed. All these results are already known for the case of a single unknown location parameter θ, but are here extended to the case in which there are additional unknown parameters. The paper concludes with a discussion of the applications in extreme value theory.